Subgroups — #30827581

Order 17152.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569333	C₂	1	2	yes	maximal, complemented
2	C₄×#14524066	C₂	1	1	yes	maximal, characteristic
3	#22559118	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559048	C₂	1	2	yes	maximal, complemented
5	#22559043	C₂	1	2	yes	maximal, complemented
6	#22559034	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559032	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558963	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558647	C₂	1	2	yes	maximal, complemented
10	#22558495	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524066	C₄	1	1	yes	characteristic
13	C₄×#11761629	C₂²	1	2	yes	—
14	#16863527	C₂²	1	2	yes	—
15	Q₁₆×Dic₆₇	C₄	1	1	yes	characteristic
16	C₄×#11761615	C₂²	1	2	yes	—
17	#16863481	C₄	1	2	yes	—
18	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
19	#16863478	C₄	1	1	yes	characteristic
20	#16863469	—	2	2	no	—
21	#16863466	—	2	2	no	—
22	C₈×D₂₆₈	C₂²	1	2	yes	—
23	#16863461	C₂²	1	2	yes	—
24	#16863451	—	2	2	no	—
25	#16863447	C₂²	1	1	yes	characteristic
26	#16863433	C₄	1	1	yes	characteristic
27	#16863418	C₂²	1	2	yes	—
28	C₄⋊C₈×D₆₇	—	2	2	no	—
29	#16863391	C₄	1	2	yes	—
30	C₄×#11761611	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863147	C₂²	1	1	yes	characteristic
33	#16863146	C₂²	1	2	yes	—
34	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
35	#16863125	C₂²	1	2	yes	—
36	#16863123	C₂²	1	2	yes	—
37	#16863118	C₂²	1	1	yes	characteristic
38	#16863117	C₂²	1	1	yes	characteristic
39	#16863117	C₂²	1	1	yes	characteristic
40	#446×C₆₇	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	#334×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	#322×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#11761629	C₂×C₄	1	2	yes	—
46	#14524066	—	2	4	no	—
47	C₂×#11761615	C₂×C₄	1	2	yes	—
48	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
49	C₂×#11761611	—	2	1	no	—
50	C₂×#11761611	—	2	2	no	—
51	C₂×C₈×D₆₇	—	2	2	no	—
52	C₂×C₈×D₆₇	—	2	2	no	—
53	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
55	Q₈×Dic₆₇	C₂×C₄	1	2	yes	—
56	C₄×D₂₆₈	—	2	2	no	—
57	C₄×D₂₆₈	C₂³	1	2	yes	—
58	#14524005	—	2	2	no	—
59	#14524005	C₂×C₄	1	2	yes	—
60	#14524001	—	2	2	no	—
61	#14524001	C₂×C₄	1	2	yes	—
62	C₄²×D₆₇	—	2	2	no	—
63	C₄²×D₆₇	C₂³	1	1	yes	characteristic
64	#14523977	C₂×C₄	1	1	yes	characteristic
65	#14523974	C₂×C₄	1	2	yes	—
66	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
67	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
68	Dic₆₇⋊C₈	—	2	2	no	—
69	Dic₆₇⋊C₈	D₄	1	2	yes	—
70	#14523967	C₂×C₄	1	2	yes	—
71	#14523965	C₂×C₄	1	2	yes	—
72	#14523963	C₂×C₄	1	2	yes	—
73	(C₂×D₆₇)⋊C₈	—	2	2	no	—
74	(C₂×D₆₇)⋊C₈	—	2	2	no	—
75	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
76	#14523954	—	2	2	no	—
77	#14523953	—	2	1	no	—
78	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
85	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	—	2	2	no	—
87	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
88	#103×C₆₇	C₂×C₄	1	2	yes	—
89	#97×C₆₇	—	2	2	no	—
90	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	#11761629	—	2	4	no	—
94	#11761629	—	4	2	no	—
95	C₂×D₂₆₈	—	2	2	no	—
96	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₆₇	—	2	2	no	—
98	C₂×C₄×D₆₇	—	2	2	no	—
99	C₂×C₄×D₆₇	—	2	2	no	—
100	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
103	#11761615	—	2	4	no	—
104	D₅₃₆	M₁₆	1	4	yes	complemented
105	#11761611	—	4	2	no	—
106	#11761611	—	8	2	no	—
107	C₂×C₆₇⋊C₈	—	2	2	no	—
108	C₂×C₆₇⋊C₈	—	2	1	no	—
109	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
112	C₈×D₆₇	—	2	2	no	—
113	C₈×D₆₇	—	4	1	no	—
114	C₈×D₆₇	—	8	2	no	—
115	C₈×D₆₇	—	8	2	no	—
116	C₄⋊Dic₆₇	—	2	2	no	—
117	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
118	(C₂×D₆₇)⋊C₄	—	2	2	no	—
119	(C₂×D₆₇)⋊C₄	—	2	2	no	—
120	(C₂×D₆₇)⋊C₄	—	2	2	no	—
121	C₄×Dic₆₇	—	2	2	no	—
122	C₄×Dic₆₇	—	2	2	no	—
123	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₆₇	#41	1	1	yes	characteristic
126	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
127	Q₁₆×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
137	C₂²×D₆₇	—	2	2	no	—
138	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
139	Q₈×C₆₇	—	2	2	no	—
140	Q₈×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
150	D₂₆₈	—	2	2	no	—
151	D₂₆₈	—	4	4	no	—
152	D₂₆₈	C₂×M₁₆	1	4	yes	—
153	C₄×D₆₇	—	2	1	no	—
154	C₄×D₆₇	—	2	1	no	—
155	C₄×D₆₇	—	2	2	no	—
156	C₄×D₆₇	—	2	2	no	—
157	C₄×D₆₇	—	4	4	no	—
158	C₄×D₆₇	—	4	2	no	—
159	C₄×D₆₇	—	4	2	no	—
160	C₄×D₆₇	—	4	4	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	2	2	no	—
163	C₈×C₆₇	—	4	1	no	—
164	C₈×C₆₇	—	4	2	no	—
165	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
166	C₆₇⋊C₈	—	2	1	no	—
167	C₆₇⋊C₈	—	2	1	no	—
168	C₆₇⋊C₈	—	2	2	no	—
169	C₆₇⋊C₈	—	4	1	no	—
170	C₆₇⋊C₈	—	4	2	no	—
171	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×D₆₇	—	2	1	no	—
173	C₂×D₆₇	—	2	4	no	—
174	C₂×D₆₇	—	2	1	no	—
175	C₂×D₆₇	—	2	1	no	—
176	C₂×D₆₇	—	4	2	no	—
177	C₂×D₆₇	—	4	2	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	4	2	no	—
180	C₄×C₆₇	—	4	2	no	—
181	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
182	C₄×C₆₇	#568	1	1	yes	characteristic
183	C₄×C₆₇	#577	1	2	yes	—
184	Dic₆₇	—	2	1	no	—
185	Dic₆₇	—	2	1	no	—
186	Dic₆₇	—	4	2	no	—
187	Dic₆₇	#444	1	2	yes	—
188	#18065	—	67	1	no	maximal, Sylow
189	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
190	C₂×C₆₇	#2952	1	1	yes	characteristic
191	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
192	D₆₇	—	4	1	no	—
193	D₆₇	—	4	4	no	—
194	#2990	—	67	2	no	—
195	C₄×#135	—	67	1	no	—
196	#1593	—	67	1	no	—
197	#1593	D₆₇	1	1	yes	characteristic, complemented
198	#1582	—	67	2	no	—
199	#1582	—	67	2	no	—
200	#1580	—	67	1	no	—
201	#1580	—	67	1	no	—
202	#1566	—	67	1	no	—
203	#1489	—	67	2	no	—
204	#1451	—	67	1	no	—
205	C₆₇	#18065	1	1	yes	characteristic, complemented, Sylow
206	C₂×#135	—	67	1	no	—
207	C₄×#41	—	67	2	no	—
208	#446	—	67	2	no	—
209	#446	C₂×D₆₇	1	2	yes	—
210	C₄×Q₁₆	—	67	1	no	—
211	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
212	C₄×SD₁₆	—	67	2	no	—
213	C₄×SD₁₆	—	67	2	no	—
214	C₄×D₈	—	67	1	no	—
215	C₄×D₈	—	67	1	no	—
216	#435	—	134	2	no	—
217	#435	—	134	2	no	—
218	C₈×D₄	—	67	2	no	—
219	C₈×D₄	—	67	2	no	—
220	#432	—	134	2	no	—
221	#431	—	67	1	no	—
222	#427	—	67	1	no	—
223	#423	—	67	2	no	—
224	C₂×C₄⋊C₈	—	134	2	no	—
225	#416	—	67	2	no	—
226	C₄×M₁₆	—	67	1	no	—
227	C₂×C₄×C₈	—	67	1	no	—
228	#334	—	67	2	no	—
229	#334	—	67	1	no	—
230	#334	C₂×D₆₇	1	1	yes	characteristic
231	Q₈⋊C₈	—	67	2	no	—
232	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
233	D₄⋊C₈	—	67	2	no	—
234	D₄⋊C₈	—	67	2	no	—
235	#322	—	67	1	no	—
236	#322	—	67	1	no	—
237	#322	—	67	1	no	—
238	#322	C₂×D₆₇	1	1	yes	characteristic
239	C₂×#41	—	67	2	no	—
240	#135	—	134	4	no	—
241	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
242	C₂×SD₁₆	—	67	2	no	—
243	C₂×D₈	—	67	1	no	—
244	C₂×M₁₆	—	134	1	no	—
245	C₂×M₁₆	—	134	2	no	—
246	C₂²×C₈	—	67	1	no	—
247	C₂²×C₈	—	67	1	no	—
248	C₂²×C₈	—	134	2	no	—
249	C₂²×C₈	—	134	2	no	—
250	C₄×Q₈	—	67	2	no	—
251	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
252	C₄×D₄	—	67	2	no	—
253	C₄×D₄	—	67	2	no	—
254	C₄×D₄	—	134	2	no	—
255	C₄×D₄	—	134	2	no	—
256	#117	—	67	2	no	—
257	#117	—	134	2	no	—
258	C₂×C₄²	—	67	1	no	—
259	C₂×C₄²	—	134	2	no	—
260	#107	—	67	1	no	—
261	#107	C₄×D₆₇	1	1	yes	characteristic
262	#106	—	67	2	no	—
263	C₄⋊C₈	—	2	2	no	—
264	C₄⋊C₈	—	67	2	no	—
265	C₄⋊C₈	—	67	2	no	—
266	C₄⋊C₈	—	134	2	no	—
267	C₄⋊C₈	—	134	2	no	—
268	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
269	#103	—	67	2	no	—
270	#103	C₄×D₆₇	1	2	yes	—
271	#102	—	67	2	no	—
272	#102	—	67	2	no	—
273	C₂²⋊C₈	—	67	2	no	—
274	C₂²⋊C₈	—	134	2	no	—
275	C₂²⋊C₈	—	134	2	no	—
276	#97	—	2	2	no	—
277	#97	—	134	2	no	—
278	#97	—	134	1	no	—
279	C₄×C₈	—	67	1	no	—
280	C₄×C₈	—	67	1	no	—
281	C₄×C₈	—	67	1	no	—
282	C₄×C₈	—	67	1	no	—
283	C₄×C₈	—	67	1	no	—
284	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
287	#41	—	268	2	no	—
288	#41	—	134	4	no	—
289	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
290	C₂×D₄	—	67	2	no	—
291	C₂×D₄	—	134	2	no	—
292	C₂²×C₄	—	67	1	no	—
293	C₂²×C₄	—	67	1	no	—
294	C₂²×C₄	—	67	1	no	—
295	C₂²×C₄	—	134	2	no	—
296	C₂²×C₄	—	134	2	no	—
297	C₂²×C₄	—	134	2	no	—
298	Q₁₆	—	2	2	no	—
299	SD₁₆	—	134	4	no	—
300	D₈	—	67	4	no	—
301	M₁₆	—	268	2	no	—
302	M₁₆	—	536	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	268	1	no	—
306	C₂×C₈	—	536	2	no	—
307	C₂×C₈	—	536	2	no	—
308	C₂×C₈	—	67	1	no	—
309	C₂×C₈	—	67	1	no	—
310	C₂×C₈	—	67	2	no	—
311	C₂×C₈	—	134	2	no	—
312	C₂×C₈	—	134	1	no	—
313	C₂×C₈	—	134	2	no	—
314	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
315	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
316	C₂×C₈	D₄×D₆₇	1	2	yes	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	67	2	no	—
319	C₄⋊C₄	—	134	2	no	—
320	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
321	C₂²⋊C₄	—	134	2	no	—
322	C₂²⋊C₄	—	134	2	no	—
323	C₂²⋊C₄	—	134	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	67	1	no	—
326	C₄²	—	67	1	no	—
327	C₄²	—	67	1	no	—
328	C₄²	—	134	2	no	—
329	C₄²	—	134	2	no	—
330	C₄²	C₂³×D₆₇	1	1	yes	characteristic
331	C₂³	—	67	1	no	—
332	C₂³	—	134	2	no	—
333	Q₈	—	2	2	no	—
334	Q₈	—	2	2	no	—
335	D₄	—	268	4	no	—
336	D₄	—	67	4	no	—
337	D₄	—	134	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	268	4	no	—
341	C₂×C₄	—	268	4	no	—
342	C₂×C₄	—	268	2	no	—
343	C₂×C₄	—	268	2	no	—
344	C₂×C₄	—	67	1	no	—
345	C₂×C₄	—	67	1	no	—
346	C₂×C₄	—	67	1	no	—
347	C₂×C₄	—	134	2	no	—
348	C₂×C₄	—	134	2	no	—
349	C₂×C₄	—	134	1	no	—
350	C₂×C₄	—	134	1	no	—
351	C₂×C₄	—	134	2	no	—
352	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
354	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	2	2	no	—
357	C₈	—	4	1	no	—
358	C₈	—	4	2	no	—
359	C₈	—	268	1	no	—
360	C₈	—	268	2	no	—
361	C₈	—	134	1	no	—
362	C₈	—	134	1	no	—
363	C₈	—	134	2	no	—
364	C₈	M₁₆×D₆₇	1	2	yes	—
365	C₂²	—	268	2	no	—
366	C₂²	—	268	2	no	—
367	C₂²	—	134	1	no	—
368	C₂²	—	134	1	no	—
369	C₂²	—	134	4	no	—
370	C₂²	—	134	1	no	—
371	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	4	2	no	—
374	C₄	—	4	2	no	—
375	C₄	—	268	2	no	—
376	C₄	—	67	2	no	—
377	C₄	—	134	1	no	—
378	C₄	—	134	1	no	—
379	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
380	C₄	#16864161	1	1	yes	characteristic
381	C₄	#16864205	1	2	yes	—
382	C₂	—	268	4	no	—
383	C₂	—	268	1	no	—
384	C₂	#22568945	1	1	yes	characteristic
385	C₂	#22569089	1	1	yes	characteristic
386	C₂	#444×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1593	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524066	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1593	D₆₇
hypocenter	C₆₇	#18065
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22569333	C₂	semidirect
3. #22559118	C₂	semidirect
4. #22559048	C₂	semidirect
5. #22559043	C₂	semidirect
6. #22559034	C₂	semidirect
7. #22559032	C₂	semidirect
8. #22558963	C₂	semidirect
9. #22558647	C₂	semidirect
10. #22558495	C₂	semidirect
11. C₆₇×#1593	C₂	semidirect
104. D₅₃₆	M₁₆	semidirect
197. #1593	D₆₇	semidirect
205. C₆₇	#18065	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22569333	2
C₄×#14524066	2
#22559118	2
#22559048	2
#22559043	2
#22559034	2
#22559032	2
#22558963	2
#22558647	2
#22558495	2
C₆₇×#1593	2
#18065	67